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## Homework Statement

Prove that the [itex]X^2[/itex] distribution is

*stochastically increasing*in its degrees of freedom; that is if [itex]p>q[/itex], then for any [itex]a[/itex], [itex]P(X^2_{p} > a) \geq P(X^2_{q} > a)[/itex], with strict inequality for some [itex]a[/itex].

## Homework Equations

1.[itex](n-1)S^2/\sigma^2 \sim X^2_{n-1}[/itex]

2.The

**Chi squared**(p)

[itex]f(x|p)= \frac 1{\Gamma(p/2) 2^{p/2}}x^{(p/2) - 1}e^{-x/2} [/itex]

## The Attempt at a Solution

Since [itex]p>q[/itex], this implies [itex]\forall a, \frac{\sigma^2 a}{p}< \frac{\sigma^2 a}{q} [/itex].

Also, [itex]X^2_{k} \sim kS^2/\sigma^2 [/itex].

Therefore [itex] \forall a, P(X^2_{p}>a) = P(S^2 > \sigma^2 a/p) \geq P(S^2 > \sigma^2 a/q) = P(X^2_{q}>a) [/itex].

If [itex] a>0[/itex], we observe strict inequality, as the support of [itex]S^2[/itex] is [itex][0,\infty) [/itex]...

What do you think? If I am going in the wrong direction, please steer me in the right one.